Atomic Physics Question 2
Question 2 - 2024 (01 Feb Shift 2)
A particular hydrogen - like ion emits the radiation of frequency $3 \times 10^{15} \mathrm{~Hz}$ when it makes transition from $n=2$ to $n=1$. The frequency of radiation emitted in transition from $\mathrm{n}=3$ to $\mathrm{n}=1$ is $\frac{x}{9} \times 10^{15} \mathrm{~Hz}$, when $\mathrm{x}=$
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Answer: (32)
Solution:
$E=-13.6 z^{2}\left(\frac{1}{n_{i}^{2}}-\frac{1}{n_{f}^{2}}\right)$
$\mathrm{E}=\mathrm{C}\left(\frac{1}{\mathrm{n}{\mathrm{f}}^{2}}-\frac{1}{\mathrm{n}{\mathrm{i}}^{2}}\right)$
$h v=C\left[\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right]$
$\frac{v_{1}}{v_{2}}=\frac{\left[\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right]{2-1}}{\left[\frac{1}{n{f}^{2}}-\frac{1}{n_{i}^{2}}\right]_{3-1}}$
$=\frac{\left[\frac{1}{1}-\frac{1}{4}\right]}{\left[\frac{1}{1}-\frac{1}{9}\right]}=\frac{3 / 4}{8 / 9}$
$=\frac{3}{4} \times \frac{9}{8}$
$\frac{v_{1}}{v_{2}}=\frac{27}{32}$
$v_{2}=\frac{32}{27} v_{1}=\frac{32}{27} \times 3 \times 10^{15} \mathrm{~Hz}=\frac{32}{9} \times 10^{15} \mathrm{~Hz}$
